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Five-fold Symmetry in AI
Ian Beardsley (Sept 22, 2021)!
Physics, University of Oregon
Genesis Project California 2021
⋅
⋅
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Abstract
We show that the artificial intelligence elements take the form of mathematical theorems,
suggesting that AI can be taken as a mathematical construct in the physical nature of the
elements from which it is made.!
1. Introduction!
To make artificial intelligence (AI) we need semiconductors, like diodes and transistors. To
make semi conductors we need to dope Silicon Si 4- with a group 13 doping agent to have
positive silicon such as with boron B 3- or with a group 15 doping agent like phosphorus P 5-
to have negative type silicon. Or we can dope germanium Ge 4- with a group 13 doping agent
like gallium Ga 3- for positive type germanium or with a group 15 doping agent like arsenic As
5- to have negative type silicon. We connect the negative with the positive to have a
semiconductor, meaning a current can run through it in only one direction. !
We pull these AI elements out of the periodic table of the elements to make an AI periodic
table:!
We now notice we can make a 3 by 3 matrix of it, which lends itself to to the curl of a vector
field, by including biological elements carbon C (above Si):!
=!
=!
=!
!
i
j
k
∂
∂x
∂
∂y
∂
∂z
(C ⋅ P)y (Si ⋅ Ga)z (Ge ⋅ As)y
(Ge ⋅ As − Si ⋅ Ga)
i + (C ⋅ P)
k
[
(72.64)(74.92) − (28.09)(69.72)
]
i +
[
(12.01)(30.97)
]
k
3,482
(
g
mol
)
2
i + 372
(
g
mol
)
2
k
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Let us dot this with and take the double integral over Si to Ge
over both variable sets:!
=!
=!
=!
=!
!
!
Now let us take the harmonic mean between Si and Ge. It is!
!
And the arithmetic mean between them:!
!
We see the value of 44.3 g/mol is somewhere between the harmonic and arithmetic mean.
Perhaps it is the geometric mean…!
!
Thus we can say…!
(
zd ydz
i + yd xd y
k
)
d xd y
∫
Ge
Si
∫
Ge
Si
(
3,483
(
g
mol
)
2
i + 372
(
g
mol
)
2
k
)
⋅
(
zd ydz
i + yd xd y
k
)
∫
Ge
Si
∫
Ge
Si
(
3,483
(
g
mol
)
2
⋅ zdzd y + 372
(
g
mol
)
2
⋅ yd xd y
)
∫
Ge
Si
3,483
(
(72.64 − 28.09)
2
2
)
dy +
∫
Ge
Si
372y ⋅ (72.64 − 28.09)d y
3456359
(
g
mol
)
4
(72.64 − 28.09) + 16573
(
g
mol
)
3
(
(72.64 − 28.09)
2
2
)
170427030.8
(
g
mol
)
5
5
170427030.8 = 44.3g/mol
2SiGe
Si + Ge
= 40.5g/m ol
Si + Ge
2
= 50.365g/mol
SiGe = 45g/m ol
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!
!
Which like Stoke’s Theorem in that it relates an integral of a flux over a surface to path integral.
The expression on the right-hand side of the equation is the geometric mean between Si and
Ge. This integral can better be represented with product calculus:!
!
Where and and n=2. If we we say the arithmetic mean is A, and the harmonic
mean is H, the geometric mean G…!
!
This is!
!
This is quite interesting because!
!
!
!
I say interesting because we can write all three of these as one equation, the f-mean:!
!
The harmonic mean and the arithmetic mean are special cases of the power-mean which is the
case when , the harmonic mean when p=-1, and the arithmetic mean when p=1.!
u = (CP ⋅ y, SiGa ⋅ z, Ga As ⋅ y)
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a = exp
(
1
Ge − Si
∫
Ge
Si
ln(x)d x
)
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a =
n
n
∏
i=1
x
i
x
1
= Si
x
2
= Ge
A + H
2
= 45.4325 ≈ G
Si
2
+ 6SiGe + Ge
2
4(Si + Ge)
≈ SiGe
H(a, b) =
1
1
b − a
∫
b
a
dx
x
A(a, b) =
1
b − a
∫
b
a
xd x
G(a, b) = exp
(
1
Ge − Si
∫
b
a
ln(x)d x
)
M
f
(x
1
, …x
n
) = f
−1
(
1
n
n
∑
i=1
f (x
i
)
)
f (x) = x
p
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But what is interesting to me is that to get the geometric mean from the f-mean we have to
change the function f(x) to f(x)=ln(x). This is when it becomes simpler to express the geometric
mean in terms of product notation:!
!
And this is precisely interesting to me because five-fold geometry does a similar thing. We have
a five-fold expression in our AI equation we arrived at:!
!
In that we take the fifth root of the double integral on left. This makes me thing of how we can
tile a surface with regular polygons the 3-sided (triangle), 4 sided (square), and 6-sided (regular
hexagon) but five pops out and the pentagon requires another shape added in to tile a surface
without leaving gaps as a so-called Archimedean tessellator, the equilateral triangle, square,
and regular hexagon are the regular tessellators. However, if you are working with solids, there
are five regular solids and they all tile to close off a space, using triangles, for example the
tetrahedron, or squares (the cube), and yes the regular pentagon in the dodecahedron.!
See illustration on next page…#
M
0
(x
1
, . . x
n
) =
n
n
∏
i=1
x
i
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a =
n
n
∏
i=1
x
i
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#
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It was the Russian scientist Shubnikov who noticed that five-fold symmetry is more
characteristic of life while six-fold symmetry is more characteristic of the physical. He wrote:!
As to the alive organisms, we have not for them theory, which could answer the question what
kinds of symmetry are compatible or incompatible to existence of living material. But we can
note here that remarkable fact that among the representations of the alive nature the
pentagonal symmetry meets more often.
I think from experience and observation you will find this as true if you pay close attention to
Nature. You will find if you look at flowers every now and then you will find six petals around its
center, or sometimes as with a rose perhaps near a hundred petals, but most often you will find
there are five petals around the center of a flower. As well, even in the rose, with near a
hundred petals, they spiral in as a golden spiral, which is built of ratios of the golden ratio (
and use patterns of Fibonacci numbers. The successive ratios between terms in the Fibonacci
sequence converge on at infinity and the golden ratio is derived from pentagonal symmetry
in that if you draw in the chord of a regular pentagon, the ratio of it to its side is . And indeed
the human has two legs, two arms and a head adding up to five, or two eyes, and a nose and a
mouth adding up to five. Or, five fingers, or five toes on each hand or each foot. But for the
physical like a snowflake, there are six points that form around it giving it hexagonal symmetry.
The starfish has five arms.!
In looking at life we notice it is based on carbon which is in group 14 of the periodic table of
the elements just like semiconductor elements silicon and germanium. It is because of this that
carbon works because it means has 4 valence electrons, meaning it can form long chains with
hydrogen making organic matter the hydrocarbons, utilizing oxygen (O), nitrogen (N),
phosphorus (P), and sulfur (S). Life does not seem to be based on silicon, though, even though
it has 4 valence electrons as well because while carbon can combine with hydrogen to make
hydrocarbons such as CH4, or combine with O, N, H to make the most simple organic
compound isocyanic acid HNCO which binds H-N=C=O, silicon in the presence of oxygen
forms glass SiO2 so easily that it can not combine with the H, N, C, O, P, and S readily with
each equally so as to form functional hydrocarbons.!
It is at this point that I would like to note that carbon is element six in the periodic table giving it
6 protons, and since its molar mass is 12.01, it has 6 neutrons. It so happens that closest
packing of equal radius spheres in the plane like protons, and neutrons is six-around one or
hexagonal symmetry. As Buckminster Fuller constructed his geometry in Synergetics, he
outlined his discovery that equal-radius spheres pack in the form of what he called the vector
equilibrium, which is the cuboctahedron, which he demonstrated was the most transformable
construct and as such becomes pivotal to his Synergetics,!
I would like to suggest in light of this that since carbon has six protons and six electrons, with
the six protons determining its number of electrons (6 to be neutral) giving it four valence
electrons in its outer shell for combining with other elements (the outer shell is four and wants
four to complete an octet, such as four hydrogens each H+, that though life more often meets
with pentagonal symmetry, and here we see carbon meets with six-around-one in the plane, or
twelve-around-one in space as the vector equilibrium, or six-fold symmetry, it is because life is
built out of the physical, like carbon to make the biological, characteristic of pentagonal
symmetry. And it is here I suggest that life animates out of a dynamic structuring of the
physical (inanimate). See illustration on the next page…#
Φ)
Φ
Φ
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#
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Indeed we see life could be the interplay between 3, 4, 5, 6 as structured in Buckminster
Fuller’s Synergetics. For instance the vector equilibrium (cuboctahedron) is made of equilateral
triangles and squares, the regular tessellators. With eight triangles and six squares. All of this
speaks respectively of NH3 (ammonia, believed to have contributed to making the amino acids
the building blocks of life) which is three hydrogens around a Nitrogen, CH4 (methane, believed
to have contributed to the formation of amino acids in primordial earth as well) the eight
triangles in the cuboctahedron representing the combination of elements such that they
complete an octet, and its six squares, the six protons, six neutrons, and six electrons of
carbon.!
With all said here so far, it might be said that understanding life and its origins can be
understood by looking at artificial intelligence.!
2. The Mathematical Nature of the Means
Let us return to the geometric mean becoming a different function in the f-mean. We have:!
!
!
p=1 yields:!
!
Is the arithmetic mean between x1 and x2. Now take p=-1:!
=!
!
Is the harmonic mean between x1 and x2. Now we try p=0 hoping to get the geometric mean…!
=!
=!
M
f
= f
−1
(
1
n
n
∑
i=1
f (x
i
)
)
M
f
(x
1
, x
2
) = f
−1
(
1
n
2
∑
i=1
x
p
i
)
p
M
f
(x
1
, x
2
) =
(
1
2
x
1
+
1
2
x
2
)
=
x
1
+ x
2
2
M
f
(x
1
, x
2
) =
(
1
2
2
∑
i=1
x
−1
i
)
−1
2
1
x
1
+
1
x
2
=
2x
1
x
2
x
1
+ x
2
M
f
(x
1
, x
2
) =
(
1
2
2
∑
i=1
x
i
0
)
1
0
(
1
2
2
∑
i=1
1
)
1
0
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!
So for we can’t make sense and we have to search for a function that will produce
the geometric mean in the f-mean. It is ln(x). This is interesting because the natural log of x was
created to settle the following conundrum:!
!
This is where we need to create the natural logarithm function so we can have a solution to
such an integral and, we have!
!
Where!
!
!
Let us return to our . It is not a sum!
!
But is a product!
=!
!
What this says is that what is important is not the values of data points in an experiment, not
the ’s but the i’s themselves, the number of the data point. Like the one in measurement 1,
the 2 in measurement 2. Never mind that measurement 1 might equal 2.3 grams, measurement
3 might equal 0.5 grams, the important thing is the 1/2 outside the parenthesis because we are
taking, which is is always 1. This is how reality never has any meaning: we
just change to f(x)= ln(x), which is the equivalent of writing!
(
1
2
)
∞
f (x) = x
p
∫
d x
x
=
∫
x
−1
d x =
x
−1+1
0
=
x
0
0
∫
d x
x
= ln(x) + C
ln(x) = log
e
(x)
e = 2.718…
(
1
2
)
∞
1
2
∞
∑
i=1
i
i
=
1
2
(
1
1
+
2
2
+
3
3
+ …
)
M
0
(x
1
, x
2
, x
3
…x
n
) =
1
2
∞
n
∏
i=1
i
i
1
2
(
1
1
⋅
2
2
⋅
3
3
⋅ …
)
x
i
i /i
1/1,2/2,3/3,...
f (x) = x
p
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!
Thus it is the experience itself that counts, we find !
!
If we say e=2.718…!
Let us see the derivations and plots…#
G =
n
n
∏
i=1
x
i
∫
d x
x
= ln(x) + C
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!
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!
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!
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!
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3. Conclusion
While we have the AI BioMatrix!
!
We can form another 3X3 matrix we will call the electronics matrix:!
!
We notice the middle column comprises the most used ductile, malleable, conductive metals
for making electrical wire Cu, Ag, Au,… just like the middle column in the AI BioMatrix contains
C, Si, Ge the semiconductor elements most used in transistor technology Si and Ge and the
core element of biological life C. Grouping the conductors and semiconductors together we
have Si, Ge, Cu, Ag, and Au which make five elements (n=5) which we need to remove the
fifth-root sign in our equation!
!
Thus, we have:!
!
!
=(28.09)(72.64)(12.01)(107.87)(196.97)=!
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a =
n
n
∏
i=1
x
i
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a =
5
∏
i=1
x
i
5
∏
i=1
x
i
= Si ⋅ Ge ⋅ Cu ⋅ Ag ⋅ Au
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!
Taking the fifth root we have 55.375 g/mol which is close to iron (Fe=55.84 g/mol). This is
(55.375)/(55.85)=99% accuracy.!
But since we have:!
!
We take the ratio and have!
!
Almost exactly 3 which is the ratio of the perimeter of regular hexagon to its diameter used to
estimate pi in ancient times by inscribing it in a circle:!
!
Perimeter=6!
Diameter=2!
6/2=3!
!
520680539
r
5
mol
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a = 170427030.8(g/mol )
5
520680539
170427030.8
= 3.055
π = 3.141...
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The Author!
